# Dictionary Definition

indistinguishability n : exact sameness; "they
shared an identity of interests" [syn: identity, identicalness]

# User Contributed Dictionary

## English

### Noun

- The state of being indistinguishable

### Quotations

- 1991 Earl H. Tilford Jnr. - Setup: what the Air Force did in
Vietnam and why
- At times Air Force and CAS efforts blended to the point of indistinguishability.

- 1996 Bose-Einstein Condensation: An Introduction
- But due to the indistinguishability of the particles in the sample, it describes a decay mechanism that occurs for all N atoms simultaneously.

# Extensive Definition

Identical particles, or indistinguishable
particles, are particles that cannot be distinguished from one
another, even in principle. Species of identical particles include
elementary
particles such as electrons, as well as composite
microscopic particles such as atoms and molecules.

There are two main categories of identical
particles: bosons, which
can share quantum
states, and fermions, which are forbidden
from sharing quantum states (this property of fermions is known as
the Pauli
exclusion principle.) Examples of bosons are photons, gluons, phonons, and helium-4 atoms.
Examples of fermions are electrons, neutrinos, quarks, protons and neutrons, and helium-3
atoms.

The fact that particles can be identical has
important consequences in statistical
mechanics. Calculations in statistical mechanics rely on
probabilistic arguments, which are sensitive to whether or not the
objects being studied are identical. As a result, identical
particles exhibit markedly different statistical behavior from
distinguishable particles. For example, the indistinguishability of
particles has been proposed as a solution to Gibbs' mixing
paradox.

## Distinguishing between particles

There are two ways in which one might distinguish
between particles. The first method relies on differences in the
particles' intrinsic physical properties, such as mass, electric
charge, and spin. If
differences exist, we can distinguish between the particles by
measuring the relevant properties. However, it is an empirical fact
that microscopic particles of the same species have completely
equivalent physical properties. For instance, every electron in the
universe has exactly the same electric charge; this is why we can
speak of such a thing as "the charge
of the electron".

Even if the particles have equivalent physical
properties, there remains a second method for distinguishing
between particles, which is to track the trajectory of each
particle. As long as we can measure the position of each particle
with infinite precision (even when the particles collide), there
would be no ambiguity about which particle is which.

The problem with this approach is that it
contradicts the principles of quantum
mechanics. According to quantum theory, the particles do not
possess definite positions during the periods between measurements.
Instead, they are governed by wavefunctions that give the
probability of finding a particle at each position. As time passes,
the wavefunctions tend to spread out and overlap. Once this
happens, it becomes impossible to determine, in a subsequent
measurement, which of the particle positions correspond to those
measured earlier. The particles are then said to be
indistinguishable.

## Quantum mechanical description of identical particles

### Symmetrical and antisymmetrical states

We will now make the above discussion concrete,
using the formalism developed in the article on the
mathematical formulation of quantum mechanics.

For simplicity, consider a system composed of two
identical particles. As the particles possess equivalent physical
properties, their state vectors occupy mathematically identical
Hilbert
spaces. If we denote the Hilbert space of a single particle as
H, then the Hilbert space of the combined system is formed by the
tensor
product H \otimes H.

Let n denote a complete set of (discrete) quantum
numbers for specifying single-particle states (for example, for the
particle
in a box problem we can take n to be the quantized wave vector
of the wavefunction.) Suppose that one particle is in the state n1,
and another is in the state n2. What is the quantum state of the
system? We might guess that it is

- |n_1\rang |n_2\rang

which is simply the canonical way of constructing
a basis for a tensor product space from the individual spaces.
However, this expression implies that we can identify the particle
with n1 as "particle 1" and the particle with n2 as "particle 2",
which conflicts with the ideas about indistinguishability discussed
earlier.

Actually, it is an empirical fact that identical
particles occupy special types of multi-particle states, called
symmetric states and antisymmetric states. Symmetric states have
the form

- |n_1, n_2; S\rang \equiv \mbox \times \bigg( |n_1\rang |n_2\rang + |n_2\rang |n_1\rang \bigg)

Antisymmetric states have the form

- |n_1, n_2; A\rang \equiv \mbox \times \bigg( |n_1\rang |n_2\rang - |n_2\rang |n_1\rang \bigg)

Note that if n1 and n2 are the same, our equation
for the antisymmetric state gives zero, which cannot be a state
vector as it cannot be normalized. In other words, in an
antisymmetric state the particles cannot occupy the same
single-particle states. This is known as the Pauli
exclusion principle, and it is the fundamental reason behind
the chemical
properties of atoms and the stability of matter.

### Exchange symmetry

The importance of symmetric and antisymmetric
states is ultimately based on empirical evidence. It appears to be
a fact of Nature that identical particles do not occupy states of a
mixed symmetry, such as

- |n_1, n_2; ?\rang = \mbox \times \bigg( |n_1\rang |n_2\rang + i |n_2\rang |n_1\rang \bigg)

There is actually an exception to this rule,
which we will discuss later. On the other hand, we can show that
the symmetric and antisymmetric states are in a sense special, by
examining a particular symmetry of the multiple-particle states
known as exchange symmetry.

Let us define a linear operator P, called the
exchange operator. When it acts on a tensor product of two state
vectors, it exchanges the values of the state vectors:

- P \bigg(|\psi\rang |\phi\rang \bigg) \equiv |\phi\rang |\psi\rang

P is both Hermitian and
unitary.
Because it is unitary, we can regard it as a
symmetry operator. We can describe this symmetry as the
symmetry under the exchange of labels attached to the particles
(i.e., to the single-particle Hilbert spaces).

Clearly, P² = 1 (the identity operator),
so the eigenvalues of
P are +1 and −1. The corresponding eigenvectors are the
symmetric and antisymmetric states:

- P|n_1, n_2; S\rang = + |n_1, n_2; S\rang
- P|n_1, n_2; A\rang = - |n_1, n_2; A\rang

In other words, symmetric and antisymmetric
states are essentially unchanged under the exchange of particle
labels: they are only multiplied by a factor of +1 or −1,
rather than being "rotated" somewhere else in the Hilbert space.
This indicates that the particle labels have no physical meaning,
in agreement with our earlier discussion on
indistinguishability.

We have mentioned that P is Hermitian. As a
result, it can be regarded as an observable of the system, which
means that we can, in principle, perform a measurement to find out
if a state is symmetric or antisymmetric. Furthermore, the
equivalence of the particles indicates that the
Hamiltonian can be written in a symmetrical form, such as

- H = \frac + \frac + U(|x_1 - x_2|) + V(x_1) + V(x_2)

It is possible to show that such Hamiltonians
satisfy the commutation
relation

- \left[P, H\right] = 0

According to the Heisenberg
equation, this means that the value of P is a constant of
motion. If the quantum state is initially symmetric
(antisymmetric), it will remain symmetric (antisymmetric) as the
system evolves. Mathematically, this says that the state vector is
confined to one of the two eigenspaces of P, and is not allowed to
range over the entire Hilbert space. Thus, we might as well treat
that eigenspace as the actual Hilbert space of the system. This is
the idea behind the definition of Fock
space.

### Fermions and bosons

The choice of symmetry or antisymmetry is
determined by the species of particle. For example, we must always
use symmetric states when describing photons or helium-4 atoms, and antisymmetric
states when describing electrons or protons.

Particles which exhibit symmetric states are
called bosons. As we will
see, the nature of symmetric states has important consequences for
the statistical properties of systems composed of many identical
bosons. These statistical properties are described as Bose-Einstein
statistics.

Particles which exhibit antisymmetric states are
called fermions. As we
have seen, antisymmetry gives rise to the Pauli
exclusion principle, which forbids identical fermions from
sharing the same quantum state. Systems of many identical fermions
are described by Fermi-Dirac
statistics.

Parastatistics
are also possible.

In certain two-dimensional systems, mixed
symmetry can occur. These exotic particles are known as anyons, and they obey fractional
statistics. Experimental evidence for the existence of anyons
exists in the fractional
quantum Hall effect, a phenomenon observed in the
two-dimensional electron gases that form the inversion layer of
MOSFETs.
There is another type of statistic, known as braid
statistics, which are associated with particles known as
plektons.

The spin-statistics
theorem relates the exchange symmetry of identical particles to
their spin. It
states that bosons have integer spin, and fermions have
half-integer spin. Anyons possess fractional spin.

### N particles

The above discussion generalizes readily to the
case of N particles. Suppose we have N particles with quantum
numbers n1, n2, ..., nN. If the particles are bosons, they occupy a
totally symmetric state, which is symmetric under the exchange of
any two particle labels:

- |n_1 n_2 \cdots n_N; S\rang = \sqrt \sum_p |n_\rang |n_\rang \cdots |n_\rang

Here, the sum is taken over all possible permutations p acting on N
elements. The square root on the right hand side is a normalizing
constant. The quantity Nj stands for the number of times each
of the single-particle states appears in the N-particle
state.

In the same vein, fermions occupy totally
antisymmetric states:

- |n_1 n_2 \cdots n_N; A\rang = \frac \sum_p \mathrm(p) |n_\rang |n_\rang \cdots |n_\rang\

Here, sgn(p) is the signature
of each permutation (i.e. +1 if p is composed of an even number of
transpositions, and −1 if odd.) Note that we have omitted
the ΠjNj term, because each single-particle state can appear
only once in a fermionic state.

These states have been normalized so that

- \lang n_1 n_2 \cdots n_N; S | n_1 n_2 \cdots n_N; S\rang = 1, \qquad \lang n_1 n_2 \cdots n_N; A | n_1 n_2 \cdots n_N; A\rang = 1.

### Measurements of identical particles

Suppose we have a system of N bosons (fermions)
in the symmetric (antisymmetric) state

- |n_1 n_2 \cdots n_N; S/A \rang

and we perform a measurement of some other set of
discrete observables, m. In general, this would yield some result
m1 for one particle, m2 for another particle, and so forth. If the
particles are bosons (fermions), the state after the measurement
must remain symmetric (antisymmetric), i.e.

- |m_1 m_2 \cdots m_N; S/A \rang

The probability of obtaining a particular result
for the m measurement is

- P_(n_1, \cdots n_N \rightarrow m_1, \cdots m_N) \equiv \bigg|\lang m_1 \cdots m_N; S/A \,|\, n_1 \cdots n_N; S/A \rang \bigg|^2

We can show that

- \sum_ P_(n_1, \cdots n_N \rightarrow m_1, \cdots m_N) = 1

which verifies that the total probability is 1.
Note that we have to restrict the sum to ordered values of m1, ...,
mN to ensure that we do not count each multi-particle state more
than once.

### Wavefunction representation

So far, we have worked with discrete observables.
We will now extend the discussion to continuous observables, such
as the position
x.

Recall that an eigenstate of a continuous
observable represents an infinitesimal range of values of the
observable, not a single value as with discrete observables. For
instance, if a particle is in a state |ψ>, the
probability of finding it in a region of volume d³x
surrounding some position x is

- |\lang x | \psi \rang|^2 \; d^3 x

As a result, the continuous eigenstates |x>
are normalized to the delta
function instead of unity:

- \lang x | x' \rang = \delta^3 (x - x')

We can construct symmetric and antisymmetric
multi-particle states out of continuous eigenstates in the same way
as before. However, it is customary to use a different normalizing
constant:

- |x_1 x_2 \cdots x_N; S\rang = \frac \sum_p |x_\rang |x_\rang \cdots |x_\rang
- |x_1 x_2 \cdots x_N; A\rang = \frac \sum_p \mathrm(p) |x_\rang |x_\rang \cdots |x_\rang

We can then write a many-body wavefunction,

\Psi^_ (x_1, x_2, \cdots x_N) \equiv \lang x_1
x_2 \cdots x_N; S | n_1 n_2 \cdots n_N; S \rang

= \sqrt \sum_p \psi_(x_1) \psi_(x_2) \cdots
\psi_(x_N)

\Psi^_ (x_1, x_2, \cdots x_N) \equiv \lang x_1
x_2 \cdots x_N; A | n_1 n_2 \cdots n_N; A \rang

= \frac \sum_p \mathrm(p) \psi_(x_1) \psi_(x_2)
\cdots \psi_(x_N)

where the single-particle wavefunctions are
defined, as usual, by

- \psi_n(x) \equiv \lang x | n \rang

The most important property of these
wavefunctions is that exchanging any two of the coordinate
variables changes the wavefunction by only a plus or minus sign.
This is the manifestation of symmetry and antisymmetry in the
wavefunction representation:

\Psi^_ (\cdots x_i \cdots x_j\cdots) = \Psi^_
(\cdots x_j \cdots x_i \cdots)

\Psi^_ (\cdots x_i \cdots x_j\cdots) = - \Psi^_
(\cdots x_j \cdots x_i \cdots)

The many-body wavefunction has the following
significance: if the system is initially in a state with quantum
numbers n1, ..., nN, and we perform a position measurement, the
probability of finding particles in infinitesimal volumes near x1,
x2, ..., xN is

- N! \; \left|\Psi^_ (x_1, x_2, \cdots x_N) \right|^2 \; d^\!x

The factor of N! comes from our normalizing
constant, which has been chosen so that, by analogy with
single-particle wavefunctions,

- \int\!\int\!\cdots\!\int\; \left|\Psi^_ (x_1, x_2, \cdots x_N)\right|^2 d^3\!x_1 d^3\!x_2 \cdots d^3\!x_N = 1

Because each integral runs over all possible
values of x, each multi-particle state appears N! times in the
integral. In other words, the probability associated with each
event is evenly distributed across N! equivalent points in the
integral space. Because it is usually more convenient to work with
unrestricted integrals than restricted ones, we have chosen our
normalizing constant to reflect this.

Finally, it is interesting to note that that
antisymmetric wavefunction can be written as the determinant of a matrix,
known as a Slater
determinant:

- \Psi^_ (x_1, \cdots x_N)

## Statistical properties

### Statistical effects of indistinguishability

The indistinguishability of particles has a
profound effect on their statistical properties. To illustrate
this, let us consider a system of N distinguishable,
non-interacting particles. Once again, let nj denote the state
(i.e. quantum numbers) of particle j. If the particles have the
same physical properties, the njs run over the same range of
values. Let ε(n) denote the energy of a particle in state n.
As the particles do not interact, the total energy of the system is
the sum of the single-particle energies. The
partition function of the system is

- Z = \sum_ \exp\left\

where k is Boltzmann's
constant and T is the temperature. We can factorize this expression
to obtain

- Z = \xi^N

where

- \xi = \sum_n \exp\left[ - \frac \right].

If the particles are identical, this equation is
incorrect. Consider a state of the system, described by the single
particle states [n1, ..., nN]. In the equation for Z, every
possible permutation of the ns occurs once in the sum, even though
each of these permutations is describing the same multi-particle
state. We have thus over-counted the actual number of states.

If we neglect the possibility of overlapping
states, which is valid if the temperature is high, then the number
of times we count each state is approximately N!. The correct
partition function is

- Z = \frac.

Note that this "high temperature" approximation
does not distinguish between fermions and bosons.

The discrepancy in the partition functions of
distinguishable and indistinguishable particles was known as far
back as the 19th
century, before the advent of quantum mechanics. It leads to a
difficulty known as the Gibbs
paradox. Gibbs
showed that if we use the equation Z = ξN, the entropy
of a classical ideal gas
is

- S = N k \ln \left(V\right) + N f(T)

where V is the volume of the gas and f is some
function of T alone. The problem with this result is that S is not
extensive
- if we double N and V, S does not double accordingly. Such a
system does not obey the postulates of thermodynamics.

Gibbs also showed that using Z = ξN/N!
alters the result to

- S = N k \ln \left(\frac\right) + N f(T)

which is perfectly extensive. However, the reason
for this correction to the partition function remained obscure
until the discovery of quantum mechanics.

### Statistical properties of bosons and fermions

There are important differences between the
statistical behavior of bosons and fermions, which are described by
Bose-Einstein
statistics and Fermi-Dirac
statistics respectively. Roughly speaking, bosons have a
tendency to clump into the same quantum state, which underlies
phenomena such as the laser, Bose-Einstein
condensation, and superfluidity. Fermions, on
the other hand, are forbidden from sharing quantum states, giving
rise to systems such as the Fermi gas. This
is known as the Pauli Exclusion Principle, and is responsible for
much of chemistry, since the electrons in an atom (fermions)
successively fill the many states within shells
rather than all lying in the same lowest energy state.

We can illustrate the differences between the
statistical behavior of fermions, bosons, and distinguishable
particles using a system of two particles. Let us call the
particles A and B. Each particle can exist in two possible states,
labelled |0> and |1>, which have the same energy.

We let the composite system evolve in time,
interacting with a noisy environment. Because the |0> and |1>
states are energetically equivalent, neither state is favored, so
this process has the effect of randomizing the states. (This is
discussed in the article on quantum
entanglement.) After some time, the composite system will have
an equal probability of occupying each of the states available to
it. We then measure the particle states.

If A and B are distinguishable particles, then
the composite system has four distinct states:
\scriptstyle|0\rangle|0\rangle, \scriptstyle|1\rangle|1\rangle,
\scriptstyle|0\rangle|1\rangle, and \scriptstyle|1\rangle|0\rangle.
The probability of obtaining two particles in the
\scriptstyle|0\rangle state is 0.25; the probability of obtaining
two particles in the \scriptstyle|1\rangle state is 0.25; and the
probability of obtaining one particle in the |0> state and the
other in the \scriptstyle|1\rangle state is 0.5.

If A and B are identical bosons, then the
composite system has only three distinct states:
\scriptstyle|0\rangle|0\rangle, \scriptstyle|1\rangle|1\rangle, and
\scriptstyle1/\sqrt(|0\rangle|1\rangle + |1\rangle|0\rangle). When
we perform the experiment, the probability of obtaining two
particles in the |0> state is now 0.33; the probability of
obtaining two particles in the \scriptstyle|1\rangle state is 0.33;
and the probability of obtaining one particle in the |0> state
and the other in the |1> state is 0.33. Note that the
probability of finding particles in the same state is relatively
larger than in the distinguishable case. This demonstrates the
tendency of bosons to "clump."

If A and B are identical fermions, there is only
one state available to the composite system: the totally
antisymmetric state \scriptstyle1/\sqrt(|0\rangle|1\rangle -
|1\rangle|0\rangle). When we perform the experiment, we inevitably
find that one particle is in the \scriptstyle|0\rangle state and
the other is in the |1> state.

The results are summarized in Table 1:

Table 1: Statistics of two particles Particles
Both 0 Both 1 One 0 and one 1 Distinguishable 0.25 0.25 0.5 Bosons
0.33 0.33 0.33 Fermions 0 0 1

As can be seen, even a system of two particles
exhibits different statistical behaviors between distinguishable
particles, bosons, and fermions. In the articles on Fermi-Dirac
statistics and Bose-Einstein
statistics, these principles are extended to large number of
particles, with qualitatively similar results.

## The homotopy class

To understand why we have the statistics that we
do for particles, we first have to note that particles are point
localized excitations and that particles that are spacelike
separated do not interact. In a flat d-dimensional space M, at any
given time, the configuration of two identical particles can be
specified as an element of M × M. If there is no overlap
between the particles, so that they do not interact (at the same
time, we are not referring to time delayed interactions here, which
are mediated at the speed of light or slower), then we are dealing
with the space [M × M]/, the subspace with coincident
points removed. (x,y) describes the configuration with particle I
at x and particle II at y. (y,x) describes the interchanged
configuration. With identical particles, the state described by
(x,y) ought to be indistinguishable (which ISN'T the same thing as
identical!) from the state described by (y,x). Let's look at the
homotopy
class of continuous paths from (x,y) to (y,x). If M is Rd where
d\geq 3, then this homotopy class only has one element. If M is R2,
then this homotopy class has countably many elements (i.e. a
counterclockwise interchange by half a turn, a counterclockwise
interchange by one and a half turns, two and a half turns, etc, a
clockwise interchange by half a turn, etc). In particular, a
counterclockwise interchange by half a turn is NOT homotopic to a clockwise
interchange by half a turn. Lastly, if M is R, then this homotopy
class is empty. Obviously, if M is not isomorphic to Rd, we can
have more complicated homotopy classes...

What does this all mean?

Let's first look at the case
d \geq 3. The universal
covering space of [M × M]/, which is none other than
[M × M]/ itself, only has two points which are physically
indistinguishable from (x, y), namely (x, y) itself and (y, x). So,
the only permissible interchange is to swap both particles.
Performing this interchange twice gives us (x, y) back again. If
this interchange results in a multiplication by +1, then we have
Bose statistics and if this interchange results in a multiplication
by −1, we have Fermi statistics.

Now how about R2? The universal covering space of
[M × M]/ has infinitely many points which are physically
indistinguishable from (x,y). This is described by the infinite
cyclic
group generated by making a counterclockwise half-turn
interchange. Unlike the previous case, performing this interchange
twice in a row does not lead us back to the original state. So,
such an interchange can generically result in a multiplication by
exp(iθ) (its absolute value is 1 because of unitarity...). This is called
anyonic statistics. In
fact, even with two DISTINGUISHABLE particles, even though
(x, y) is now physically distinguishable from
(y, x), if we go over to the universal covering space, we
still end up with infinitely many points which are physically
indistinguishable from the original point and the interchanges are
generated by a counterclockwise rotation by one full turn which
results in a multiplication by exp(iφ). This phase factor
here is called the mutual
statistics.

As for R, even if particle I and particle II are
identical, we can always distinguish between them by the labels
"the particle on the left" and "the particle on the right". There
is no interchange symmetry here and such particles are called
plektons.

The generalization to n identical particles
doesn't give us anything qualitatively new because they are
generated from the exchanges of two identical particles.

## See also

indistinguishability in Arabic: جسيمات
متماثلة

indistinguishability in German:
Ununterscheidbare Teilchen

indistinguishability in Spanish: Partículas
idénticas

indistinguishability in French: Particules
indiscernables

indistinguishability in Galician: Partículas
Idénticas

indistinguishability in Italian: Particelle
identiche

indistinguishability in Polish: Cząstki
identyczne

indistinguishability in Russian: Тождественные
частицы

indistinguishability in Slovak: Nerozlíšiteľné
častice

indistinguishability in Swedish: Ourskiljbara
partiklar